Download Space Vector PWM and Model Predictive Control for Voltage Source

World Academy of Science, Engineering and Technology
International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering Vol:8, No:11, 2014
Space Vector PWM and Model Predictive Control for
Voltage Source Inverter Control
Irtaza M. Syed, Kaamran Raahemifar
International Science Index, Industrial and Manufacturing Engineering Vol:8, No:11, 2014 waset.org/Publication/9999743
Abstract—In this paper, we present a comparative assessment of
Space Vector Pulse Width Modulation (SVPWM) and Model
Predictive Control (MPC) for two-level three phase (2L-3P) Voltage
Source Inverter (VSI). VSI with associated system is subjected to
both control techniques and the results are compared.
Matlab/Simulink was used to model, simulate and validate the
control schemes. Findings of this study show that MPC is superior to
SVPWM in terms of total harmonic distortion (THD) and
implementation.
simultaneously to avoid uncertain output voltage. Terminals
A, B, C can be connected to three phase load or across grid
depending on off-grid or on-grid operation.
Keywords—Model Predictive Control, Space Vector Pulse Width
Modulation, Voltage Source Inverter.
T
I. INTRODUCTION
HE DC to AC power converters are known as inverters.
Inverters are mainly classified as current and voltage
source inverters. Current source inverters have DC current
source at input and use switches to adjust the output current
and frequency. On the hand, voltage source inverters (VSI)
convert DC power into AC power at desired voltage and
frequency [1]. The AC output voltage and frequency may be
fixed or variable depending on the application [2].
Semiconductor switches, such as Bipolar Junction Transistors,
Metal Oxide Semi-conductor Field Effect Transistors,
Insulated Gate Bipolar Transistors, etc., are used to control
and adjust the output parameters [3]. Anti-parallel diodes are
connected across from the switches to add a reverse current
capability [2]. VSI requires a stiff DC voltage source at input
[1] for quality operation. The input source could be the output
of a rectifier, battery, fuel cell, or photo voltaic (PV) system.
Fig. 1 shows a basic two-level three phase (2L-3P) VSI
circuit, where P is positive, N is negative terminal and O is a
null point. VSI has three legs each with two switches and two
anti-parallel diodes. For example, leg-1 has switches 1 (S1)
and 4 (S4) and diodes 1 (D1) and 4 (D4). To prevent short
circuit across terminals switches of the same leg should never
be turned on simultaneously. Therefore, intentional switching
delay (known as dead or blanking time) is introduced to
ensure the first switch is turned off before the complementary
switch of the same leg is turned on. If a short circuit happens,
the fault current rises very quickly and unless it is controlled
by proper protecting measures, it can damage the system.
Similarly, switches of the same leg are never turned off
Irtaza M. Syed is a student at the Electrical & Computer Engineering
Department, Ryerson University, Toronto, ON M5K 2K3 Canada
(corresponding author phone: 647-787-6262; e-mail: i5syed@ ryerson.ca).
Dr. Kaamran Raahemifar, is with the Electrical & Computer Engineering
Department, Ryerson University, Toronto, ON M5K 2K3 Canada, (e-mail:
[email protected]).
International Scholarly and Scientific Research & Innovation 8(11) 2014
Fig. 1 2l-3P VSI basic circuit
Though the output voltage can be controlled by adjusting
VDC, usually the more convenient method of Pulse Width
Modulation (PWM) is used. PWM adjusts pulse width by
increasing or decreasing on and off times of the switches to
control the output voltage. Conventionally, the on and off
commands are issued by comparing three phase modulating
signals (VmA, VmB, VmC) with a carrier wave (VC) (1).
0 (1)
where x=A,B,C
For example, if VmA>VC, VAN=VDC with S1 on and S4 off.
On the other hand, if VmA<VC, VAN=0 with S1 off and S4 on.
The status of switches for all three legs is shown in Table I.
S
States
P
O
TABLE I
THE STATUS OF SWITCHES IN INDIVIDUAL LEGS
Leg A
Leg B
Leg C
S1
S4
VAN
S3
S6
VBN
S5
S2
VCN
1
0
VDC
1
0
VDC
1
0
VDC
0
1
0
0
1
0
0
1
0
Since the input lines should never be shortened and the
output current should always be continuous, a voltage source
inverter can assume only 8 distinct switching states, defined
by (2) and presented in Table II.
States = n3
where n=2 for two level VSI.
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(2)
World Academy of Science, Engineering and Technology
International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering Vol:8, No:11, 2014
International Science Index, Industrial and Manufacturing Engineering Vol:8, No:11, 2014 waset.org/Publication/9999743
Legs Upper S
S1
S3
S5
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
TABLE II
VSI 8 SWITCHING STATES
Legs Lower S
S4
S6
S2
1
1
1
1
1
0
1
0
1
1
0
0
0
1
1
0
1
0
0
0
1
0
0
0
phase load voltages and line to line voltages.
VAN
VDC
VDC
VDC
VDC
0
0
0
0
Voltage
VBN
VDC
VDC
0
0
VDC
VDC
0
0
VCN
VDC
0
VDC
0
VDC
0
VDC
0
(3)
Multiple methods exist for controlling VSI. This paper
compares Space Vector Pulse Width Modulation (SVPWM)
with Model Predictive Control (MPC) for VSI. The rest of the
paper is organized into the following sections: (II) DQ
transformation, (III) SVPWM, (IV) MPC, (V) System
Performance Assessment, (VI) Results and Discussion, (VII)
Conclusions.
Individual phase voltages on the load can easily be
determined considering the fact that any switching
combination will result in a specific configuration of the VSI
circuit. Fig. 2 shows VSI circuit configuration when
S1S2S3=111.
Fig. 4 VSI circuit for S1S6S2=111
II. DQ TRANSFORMATION
In a balanced three phase system, the sum of three phase
voltage (or current) equals to zero due to the equal amplitude
and 120o phase shift (4). This renders one of the three Vs (or
Is) redundant. Therefore, if the value of two voltages (or
currents) is known, the third one can be found. This
redundancy permits conversion of the three phase (or three
frames) ABC variables to two frames αβ (or αβ0) and dq (or
dq0) variables.
0
Fig. 2 VSI circuit for S1S2S3=111
TABLE III
PHASE LOAD AND LINE TO LINE VOLTAGES
Lower leg Sw
Phase load V*
Line-Line V*
S4
S6
S2
VAO
VBO
VCO
VAB
VBC
VCA
1
1
1
0
0
0
0
0
0
1
1
0
-1/3
-1/3
2/3
0
-1
1
1
0
1
-1/3
2/3
-1/3
-1
1
0
1
0
0
-2/3
1/3
1/3
-1
0
1
0
1
1
2/3
-1/3
-1/3
1
0
-1
0
1
0
1/3
-2/3
1/3
1
-1
0
0
0
1
1/3
1/3
-2/3
0
1
-1
0
0
0
0
0
0
0
0
0
Alternatively, the VSI circuit configuration when
S1S2S3=111 can be shown as in Fig. 3. Phase A and B are
connected to the positive terminal of VDC while phase C is
connected to the negative terminal of VDC and all three of
them are connected to O (null) point through load. This
configuration puts load A and B in parallel, connected in
series with load C and the DC source. Thus load phase
voltages VAO=VBO=1/3VDC and VCO=-2/3VDC. Fig. 4 shows
the equivalent circuit for S1S6S2=111, producing VAO=2/3VDC
and VBO=VCO=-1/3VDC. It is important to note that
VAO=VBO=VCO=0 for both S1S3S5=111 (S4S6S2=000) and
S1S3S5=000 (S4S6S2=111).
Fig. 3 VSI circuit for S1S2S3=111
Line to line voltages (VAB, VBC, VCA) are obtained by
subtracting line to neutral voltage of one phase from the other.
For example, VAB is calculated by (3). Table III shows the
International Scholarly and Scientific Research & Innovation 8(11) 2014
(4)
* multiply all V's VDC
Converting the three phase control problem into two phase
control problem makes analysis, design and control of the VSI
and the associated system simple. Although αβ conversion
reduces one phase and therefore simplifies the system control,
the two dimensional frame remains sinusoidal. On the other
hand dq synchronous not only results in two phase but a DC
control problem, enabling use of PID controllers. Dq
transformation in any reference frame can be obtained by (5)
and (6). Theta (θ) is used to define the reference frame: the
reference frame is Stationary (αβ) if θ=0, Synchronous if θ=
θgrid and Arbitrary if θ is neither zero nor equal to θgrid. The αβ
reference frame is used to control VSI in both SVPWM and
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International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering Vol:8, No:11, 2014
MPC.
Equation (5) gives the transformation from ABC to dq
(direct), while (6) gives the transformation from dq to ABC
(inverse).
0
&'
*'
! cos%! ( ) cos%! ( )
. /1 2
&'
*'
+! sin%! ) sin%! )
(
(
!
0
&'
/1 2 3 cos%! ( )
cos%! *'
)
(
+!
sin%! &'
) 4 (
sin%! *'
)
(
(5)
(6)
International Science Index, Industrial and Manufacturing Engineering Vol:8, No:11, 2014 waset.org/Publication/9999743
where X=V, I, etc.
III. SVPWM
Sinusoidal PWM (SPWM), a very popular technique,
creates a relatively high harmonic distortion due to its
asymmetrical nature of the PWM switching and cannot make
use of the inverter’s supply voltage fully [3]. SVPWM
however, provides a higher voltage and lower total harmonic
distortion (THD) and is therefore, preferred.
SVPWM control strategy is based on the fact that the VSI
possible switching states are finite. The 2L-3P VSI has 6
active and 2 zero space (or voltage) vectors. The six active
vectors produce a nonzero output voltage and the two zero
vectors produce zero output voltage (Table IV).
5 6 78
(7)
Proceeding on similar lines the six non-zero voltage vectors
(V1 - V6) can assume the positions shown in Fig. 6, forming a
regular hexagon. There are six active vectors, with the area
between any two adjacent vectors defined as a sector. The
remaining two zero space vectors produce no output voltage,
and therefore they remain at origin in αβ plane defining no
sector. Space vector V is known as the reference space vector.
Note that all the 8 space vectors are stationary. The
reference space vector, V (equivalent of Vα and Vβ at angle
α), however rotates in space at angular velocity of ω (or 2πf).
At any time, V is approximated by two active space vectors
and a zero space vector. V rotates one revolution for one
complete cycle of VAB and its length corresponds to the
magnitude of VAB.
TABLE IV
VSI SPACE VECTORS
S4
1
1
1
1
0
0
0
0
S6
1
1
0
0
1
1
0
0
S2
1
0
1
0
1
0
1
0
Vector
0
2/3VDC
1/3VDC + j√3/3VDC
-1/3VDC + j√3/3VDC
-2/3VDC
-1/3VDC - j√3/3VDC
1/3VDC - j√3/3VDC
0
Vx
V0
V1
V2
V3
V4
V5
V6
V7
Fig. 6 SVPWM space vectors
To generate any magnitude of VAB at the specific angle α,
Vα and Vβ followed by V and α are determined using (5), (8)
and (9). Then the turn on durations T1, T2, and T0 are
estimated with appropriate switching sequence and switching
time for switches (S1-S6) using (10) to (12).
95 :6 8
; <=+>?
F F? √H|@|
L
@JK
√H|@|
P
@JK
@A
C< 2E
<
@B
MN
+
; +
; LM>?ON
(8)
MN
+; FR F F? F
+;O
LM>?ON
where F 1T
, n=1- 6 (sectors), and 0≤ α ≤60o [4].
Fig. 5 Space vector V6
Space vector of the three phase quantities are represented as
vectors in a two-dimensional αβ plane. For example, V6 shown
in Fig. 5 represents S1S2S3=111 (Figs. 2 & 3) or vector V6
(S4S6S2=001) in Table III. VAB, VBC, and VCA are three 120o
phase shifted line voltage vectors, represented by VAB=0,
VBC=VDC and VCA=-VDC for this case (Fig. 5) with an effective
voltage/space) vector V6 equal to 1/3VDC - j√3/3VDC. In
general,
International Scholarly and Scientific Research & Innovation 8(11) 2014
(9)
Q
(10)
(11)
(12)
There are a few options for null vector: V0 only, V7 only,
or a combination of both [5]. The type of null vector
determines the SVM technique. Based on the type of null
vector different SVPWM techniques such as right aligned,
symmetric, alternating zero vector and highest current not
switched sequences [5] are obtained. All the techniques have
almost identical DC source utilization; however, they differ in
terms of switching loss and THD. Usually the optimal choice
is 7-segment switching sequence (Fig. 7) which ensures
minimum switching per sampling period and low harmonics.
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IV. MPC
Model predictive control strategies like SVPWM take
advantage of the fact that only a finite number of possible
switching states are associated with VSI. These states are
discrete and the model of the system can be used in
association with a discrete-time model of the load to predict
the behavior of the VSI system. A selection criterion, the
objective function, is defined for selection of the optimal
future variables corresponding to the optimal future switching
state that minimizes the objective function. The objective of
the MPC scheme is to predict switching state and thus currents
that track the reference currents with minimal error. For each
sampling period predicted, the output currents are measured
and compared with the reference currents to minimize error.
Usually the sampling period is chosen as a period in which the
reference current does not change significantly; therefore, the
reference current can be considered a constant for that period.
Equation (16) is used to predict the load current for each
switching possibility. The objective function is evaluated for
each of the eight possible voltage vectors generated by the
VSI in order to calculate the future optimal value of the load
current. The optimal value of the objective function is applied
during the next sampling period. Fig. 8 shows the VSI MPC
control system.
Fig. 9 outlines step by step implementation process of MPC
for predictive control of VSI. First all the variables are
initialized and values assigned (Data). This is followed by a
process dynamic model of VSI based on the values for voltage
vectors V0 to V7 (Table IV). Then a set of control actions or
manipulated variables U(k) based on operational principles
(process experience) is developed corresponding to all
possible output states (using switching states, S4S6S2) in Table
IV. Current states of the system are measured (io-ref, io(k))
including switching states of v(k) and future outputs Y(k+1)
(future currents io(k+1) ) for k=1 to N are predicted using (16).
Fig. 8 VSI MPC control block diagram
Fig. 7 SVPWM with seven segment switching sequence
All the VSI operational requirements and constraints
outlined in Sections I and II are still valid except for the
control scheme. Similar to SVPWM, MPC is exercised in αβ
reference frame with 6 active and 2 zero vectors. In addition,
load dynamics are modeled as:
UV
W
X
Y
(13)
where R and L are load resistance and inductance,
respectively, i is the load current and v is the VSI generated
voltage vector.
Using Euler-Forward equation, the load current is
approximated by:
W
X
Z
WL[\?O>WL[O
H
From (13) and (14), we can say:
] L^ 1O P1 _H
`
(14)
Q L^O UL^O
H
`
(15)
where k=t (present) and k+1=t+1(future/predicted). Using (5),
the predicted load current in (15) can be expressed in the αβ
reference frame as:
1
6L[\?O
3
8L[\?O
0
_H
`
0
H
6L[O
+3 `
_H 4 8L[O
1
0
`
0
U6L[O
H 4 U8L[O `
(16)
International Scholarly and Scientific Research & Innovation 8(11) 2014
Objective function J (17) is used to minimize the error
between predicted output, Y(k) and the measured reference, ioref (r(k)). The optimal J(k+1) with minimum error between
predicted and measured currents is selected and the
corresponding control action, U(k), from the control action set,
U = [000 001 010 011 100 101 110 111], is applied across VSI
in the next sampling period. The output is observed and the
process is repeated. Note that in each sampling period 8
predictions are made and 8 Js are evaluated before selecting
the control action, U(k), for the next sampling period.
min bL[\?O ∑[\>?
[d? L eL[O fL[O O
(17)
V. SYSTEMS PERFORMANCE ASSESSMENT
VSI output is ideally sinusoidal and independent of load
parameters; however, the output of the practical inverter is
non-sinusoidal and contains harmonics. The performance of
VSI is therefore evaluated in terms of total harmonic distortion
(THD), Distortion Factor (DF), and lower harmonic distortion
(LOH) [3]. In addition, power loss, control complexity,
implementation ease, and cost are also considered when
evaluating VSI and the associated control plan.
The ratio of the root mean square (rms) and sum of total
harmonic components of output voltage plus rms of the
fundamental component, Vof, is called THD. It is the measure
of closeness in shape between a waveform and its Vof. The
LOH is the harmonic component with a magnitude greater
than or equal to 3% of Vof.
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Fgh ?
@ij
?/
(∑k
Md,… + O
(18)
To compare SVPWM and MPC control for 2L-3P VSI,
Advanced Energy AE 100TX, a commercially available VSI
from [6] with rated kVA= 100kW, rated VLL = 600V, rated Io=
96A, and f=60 Hz is selected.
For ma=1.0, DC link voltage is:
√2
@oo
p
Z 850 (19)
International Science Index, Industrial and Manufacturing Engineering Vol:8, No:11, 2014 waset.org/Publication/9999743
Using per unit system, the base quantities are:
v | @oo
√
wx
347 √@oo
@}
~}
96 {
3.62 €
C 2E
376.8 f=/
V ‚}
ƒ}
9.6 „g
(20)
(21)
Fig. 10 VSI system simulated
(22)
(23)
(24)
VI. SIMULATION RESULTS
Fig. 11 (top panel) shows the output of VSI with SVPWM
control and without LC filter (NF). Phase a current, Ia, is not a
pure sine wave and contains harmonics. The middle and
bottom panels of Fig. 11 show Vab and Va, respectively.
Fig. 11 Ia, Ia-rms, Vab, and Va (SVPWM-NF)
Fig. 12 shows Ia THD analysis. Ia is not a pure sine wave
and contains harmonics. Fundamental component (fc) has a
peak magnitude of 135.4 with 10.98% THD.
Fig. 9 MPC algorithm
For 1 pu load impedance, LL is 0.31 pu and RL equals to
0.95 pu, given by (25) and (26):
V` 0.31V Z 3 „g
Y` 0.31| Z 3.44 €
(25)
(26)
Finally sampling time, Ts, is set to 5µs for simulation. Fig.
10 shows the system modelled and simulated for both
SVPWM and MPC.
International Scholarly and Scientific Research & Innovation 8(11) 2014
Fig. 12 Fundamental and THD of Ia (SVPWM-NF)
Fig. 13 presents the results of the same analysis for Va and
shows that Va has fc of 490.2 with a THD equal to 51.68%.
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Fig. 13 Fundamental and THD of Va (SVPWM-NF)
Fig. 14 shows the output of VSI with MPC control and
without LC filter (NF). Note that Ia is almost a pure sine
wave, even without an LC filter.
Fig. 16 Fundamental and THD of Va (MPC-NF)
Figs. 17 and 18 show VSI output with SVPWM and MPC
controls with LC filter (YF). The inductor (L) and capacitor
(C) used are 4 mH and 370 µF respectively. Closer
observation of Fig. 17 reveals that as expected Ia-rms=96 A,
Vab-rms=600 V and Va-rms=347 V; however, there is slight
harmonics mix. Fig. 18 shows the same results for Ia-rms,
Vab-rms and Va-rms with less harmonics.
Fig. 14 Ia, Ia-rms, Vab, and Va (MPC-NF)
Figs. 15 and 16 show Ia and Va fc and THD analysis. Both
figures indicate great improvement in THD which is down
from 10.98% to 0.25% and 51.68% to 34.31% for Ia and Va
respectively.
Fig. 17 Ia, Ia-rms, Vab, and Va (SVPWM-YF)
Fig. 15 Fundamental and THD of Ia (MPC-NF)
Fig. 18 Ia, Ia-rms, Vab, and Va (MPC-YF)
Table V lists fc magnitude and THD for the waveforms in
Figs. 17 and 18.
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TABLE V
Va AND Ia FUNDAMENTAL AND HARMONICS
IaTHD(%)
Vaf
VaTHD(%)
Control
Iaf
SVPWM
135.2
2.21
489.8
3.21
MPC
134.6
1.37
503.4
0.58
VII. CONCLUSION
This paper compared SVPWM with MPC for 2L-3P VSI.
MPC was found superior in terms of THD and was easier to
implement.
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International Science Index, Industrial and Manufacturing Engineering Vol:8, No:11, 2014 waset.org/Publication/9999743
[1]
[2]
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[4]
[5]
[6]
M. D. Singh, "Power Electronics," 2nd edition, Tata McGraw-Hill,
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B. K. Boss, "Modern Power Electronics and AC Drives," Pearson
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M. H. Rashid, "Power Electronics Circuits, Devices, and Applications,"
3rd edition, Pearson Prentice Hall, 2007, ISBN:81-317-0246-4.
B. Wu, "High Power Converters and AC Drives," Wiley-IEEE Press,
2006, ISBN:0-471-731714.
Wei-Feng Zhang and Yue-Hui Yu, "Comparison of Three SVPWM
Strategies," Journal of Electronic Science and Technology of China,
Vol. 5, No. 3, 2007.
Advanced Energy, AE100TX, 100kW/600V three phase inverter
http://solarenergy.advanced-energy.com/ (accessed: May, 2014).
International Scholarly and Scientific Research & Innovation 8(11) 2014
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